Question

10 PONTOS!
Calcule a integral:
(segue anexo)

obs: resposta completa, passo a passo

Answer 1

Integral da potência

$\displaystyle\mathsf{\int~u^n~du}=\begin{cases}\mathsf{\dfrac{u^{n+1}}{n+1}~se~n\ne-1}\\\mathsf{ln|u|~se~n=-1}\end{cases}$

Potência de expoente

Racional

$\displaystyle \huge\mathsf{\sqrt[n]{a^m}=a^{\frac{m}{n}}}$

a) vamos escrever f(x) de uma forma conveniente para em seguida avaliar a integral da função.

$\mathsf{f(x)=\dfrac{\sqrt{x}}{\sqrt[3]{x}}=\dfrac{x^{\frac{1}{2}}}{x^{\frac{1}{3}}}}\\\mathsf{x^{\frac{1}{2}-\frac{1}{3}}=x^{\frac{3-2}{6}}=x^{\frac{1}{6}}}$

$\displaystyle\mathsf{\int f(x)dx=\int x^{\frac{1}{6}}dx=\dfrac{x^{\frac{1}{6}+1}}{\frac{1}{6}+1} + k}$

$\displaystyle\mathsf{\int f(x)dx=\dfrac{x^{\frac{7}{6}}}{\frac{7}{6}}}$

$\displaystyle\mathsf{\int~f(x)dx=\boxed{\boxed{\boxed{\boxed{\dfrac{6}{7}x^{\frac{7}{6}} + k}}}}}$

Integral da soma

$\displaystyle\mathsf{\int(f(x)+g(x))dx=} \\\mathsf{\int~f(x)dx+\int~g(x)dx}$

Integral da constante

$\displaystyle\mathsf{\int~k.f(x)dx=k\int~f(x)dx}$

b)

$\displaystyle\mathsf{\int~x(2x+3)dx=\int(2x^2+3x)dx}\\\displaystyle\mathsf{\int~2x^2dx+\int3xdx}\\\displaystyle\mathsf{2\int~x^2dx+3\int~x \: dx}$

$\mathsf{2\dfrac{x^{2+1}}{2+1}+3\dfrac{x^{1+1}}{1+1}+k}\\\mathsf{\dfrac{2x^3}{3}+\dfrac{3x^2}{2}+k}$